The preface, introduction, and chapter 1 of this book have been very interesting, and I can't wait to keep reading! In the meantime, here are some highlights from chapter 2:
Boaler begins chapter 2 by discussing "math wars," where the most important issue is the books the teachers will use rather than the methods. She describes a particular situation at a school district where the school had previously abandoned traditional textbooks and instead used difficult, engaging problem solving that led to student success. Some parents were not happy with this approach and worked to dismantle the program. They were successful, and now instruction in that district is more "traditional," with teachers lecturing and students working in isolation.
Boaler goes on to discuss the idea that many math classes employ passive learning - students follow and memorize methods instead of learning to inquire, ask questions, and solve problems. Students taught with passive approaches don't engage in sense making, reasoning, or thought. They don't view themselves as active problem solvers. When students passively memorize methods, they find it difficult to then apply them to new situations. This idea really made me think of myself as a student. I could figure out any problem that I knew the formula for or had memorized the method for. I was great at regurgitating information....but definitely not as good at applying concepts to new situations where I had to problem solve. That is definitely a skill that has been developing since graduation from high school.
According to data collected by PISA (Programme for International Student Assessment), students who memorize math are the lowest achieving in the world. The highest-achieving students are those who "think about the big ideas in mathematics." Hopefully, we (in the U.S.) will continue moving toward big ideas and problems solving.
Boaler shares that children begin school as natural problem solvers, and that studies have shown that they are better at solving problems before they attend math classes!
Another problem with math instruction is that many students work on math without talking, and Boaler states that this is a flawed approach. Students need to talk through solutions to see if they really understand them. She references two different mathematicians with different background who both state that talking to others and conversing with others is necessary to truly understand math ideas. To be sure students are understanding math, they need to solve complex problems and they need to talk through and explain the problems. They don't need to repeat math procedures over and over.
Talking in math leads students to understand that math is more than a set of rules and method in the books; "they realize that math is a subject that they can have their own ideas about." Mathematical discussions are an excellent resource for student understanding. When students explain and justify their work to each other, they hear one another's reasoning and are often able to understand each other better than the teacher. When verbalizing math, students have to reconstruct the ideas in their minds; when others students react to the verbalization, the ideas are reconstructed again, which leads to a deeper understanding.
In recent years, I have spent much more class time having students talk about math. They typically discuss their daily warm-ups, which are a mix of computation and shorter word problems. Other times we spend time on solving word problems related to the current unit. I have noticed some fantastic conversations - students sharing their thoughts and approaches to solving, asking one another questions, and leading each other to new ideas. Though my classroom can get a little loud at times, it's worth it! During one of my post-observations, my principal commented that during the lesson he had initially thought it was a little loud, but as he tuned in to the conversations, he was surprised and impressed to hear that the students WERE seriously talking about math and having great conversations:-)
The last section of chapter 2 is title "Learning without Reality." Boaler describes many of the math problems used in classrooms as problems that aren't realistic, requiring students to "suspend" reality. Contexts in math problems, that started being used in the 1970s and 80s are not the types of problems students would encounter in real life - the solutions they are asked to find are not the solutions they would apply in real life. This is an example she shares: "A pizza is divided into fifths for 5 friends at a party. Three of the friends eat their slices, but then 4 more friends arrive. What fractions should the remaining 2 slices be divided into?" She says that it is not representative of real life. Boaler says that the real-life solution would be that the friends would just order more pizza, or some friends would have no pizza. This section does make me think about the types of problems I give my students. While I try to use real-life situations, I think I need to find more authentic situations - something to work on!
Boaler begins chapter 2 by discussing "math wars," where the most important issue is the books the teachers will use rather than the methods. She describes a particular situation at a school district where the school had previously abandoned traditional textbooks and instead used difficult, engaging problem solving that led to student success. Some parents were not happy with this approach and worked to dismantle the program. They were successful, and now instruction in that district is more "traditional," with teachers lecturing and students working in isolation.
Boaler goes on to discuss the idea that many math classes employ passive learning - students follow and memorize methods instead of learning to inquire, ask questions, and solve problems. Students taught with passive approaches don't engage in sense making, reasoning, or thought. They don't view themselves as active problem solvers. When students passively memorize methods, they find it difficult to then apply them to new situations. This idea really made me think of myself as a student. I could figure out any problem that I knew the formula for or had memorized the method for. I was great at regurgitating information....but definitely not as good at applying concepts to new situations where I had to problem solve. That is definitely a skill that has been developing since graduation from high school. According to data collected by PISA (Programme for International Student Assessment), students who memorize math are the lowest achieving in the world. The highest-achieving students are those who "think about the big ideas in mathematics." Hopefully, we (in the U.S.) will continue moving toward big ideas and problems solving.
Boaler shares that children begin school as natural problem solvers, and that studies have shown that they are better at solving problems before they attend math classes!
Another problem with math instruction is that many students work on math without talking, and Boaler states that this is a flawed approach. Students need to talk through solutions to see if they really understand them. She references two different mathematicians with different background who both state that talking to others and conversing with others is necessary to truly understand math ideas. To be sure students are understanding math, they need to solve complex problems and they need to talk through and explain the problems. They don't need to repeat math procedures over and over.Talking in math leads students to understand that math is more than a set of rules and method in the books; "they realize that math is a subject that they can have their own ideas about." Mathematical discussions are an excellent resource for student understanding. When students explain and justify their work to each other, they hear one another's reasoning and are often able to understand each other better than the teacher. When verbalizing math, students have to reconstruct the ideas in their minds; when others students react to the verbalization, the ideas are reconstructed again, which leads to a deeper understanding.
In recent years, I have spent much more class time having students talk about math. They typically discuss their daily warm-ups, which are a mix of computation and shorter word problems. Other times we spend time on solving word problems related to the current unit. I have noticed some fantastic conversations - students sharing their thoughts and approaches to solving, asking one another questions, and leading each other to new ideas. Though my classroom can get a little loud at times, it's worth it! During one of my post-observations, my principal commented that during the lesson he had initially thought it was a little loud, but as he tuned in to the conversations, he was surprised and impressed to hear that the students WERE seriously talking about math and having great conversations:-)
The last section of chapter 2 is title "Learning without Reality." Boaler describes many of the math problems used in classrooms as problems that aren't realistic, requiring students to "suspend" reality. Contexts in math problems, that started being used in the 1970s and 80s are not the types of problems students would encounter in real life - the solutions they are asked to find are not the solutions they would apply in real life. This is an example she shares: "A pizza is divided into fifths for 5 friends at a party. Three of the friends eat their slices, but then 4 more friends arrive. What fractions should the remaining 2 slices be divided into?" She says that it is not representative of real life. Boaler says that the real-life solution would be that the friends would just order more pizza, or some friends would have no pizza. This section does make me think about the types of problems I give my students. While I try to use real-life situations, I think I need to find more authentic situations - something to work on!
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